Base field 5.5.160801.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 5x^{3} + 4x^{2} + 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} + x^{11} - 26x^{10} - 19x^{9} + 250x^{8} + 123x^{7} - 1095x^{6} - 282x^{5} + 2152x^{4} + 34x^{3} - 1603x^{2} + 175x + 305\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -w^{4} + 5w^{2} - 3]$ | $-\frac{857}{3218}e^{11} - \frac{1583}{3218}e^{10} + \frac{20297}{3218}e^{9} + \frac{16152}{1609}e^{8} - \frac{174333}{3218}e^{7} - \frac{235177}{3218}e^{6} + \frac{329084}{1609}e^{5} + \frac{719279}{3218}e^{4} - \frac{517564}{1609}e^{3} - \frac{812571}{3218}e^{2} + \frac{505287}{3218}e + \frac{130214}{1609}$ |
| 9 | $[9, 3, -w^{4} + w^{3} + 5w^{2} - 3w - 2]$ | $-\frac{3013}{6436}e^{11} - \frac{5235}{6436}e^{10} + \frac{74213}{6436}e^{9} + \frac{28228}{1609}e^{8} - \frac{661407}{6436}e^{7} - \frac{876491}{6436}e^{6} + \frac{1282433}{3218}e^{5} + \frac{2853611}{6436}e^{4} - \frac{2016783}{3218}e^{3} - \frac{3325395}{6436}e^{2} + \frac{1936155}{6436}e + \frac{533641}{3218}$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $\phantom{-}1$ |
| 17 | $[17, 17, w^{4} - w^{3} - 5w^{2} + 3w + 1]$ | $-\frac{483}{6436}e^{11} - \frac{262}{1609}e^{10} + \frac{5193}{3218}e^{9} + \frac{20127}{6436}e^{8} - \frac{79497}{6436}e^{7} - \frac{34233}{1609}e^{6} + \frac{266929}{6436}e^{5} + \frac{399869}{6436}e^{4} - \frac{394677}{6436}e^{3} - \frac{483261}{6436}e^{2} + \frac{99395}{3218}e + \frac{160393}{6436}$ |
| 19 | $[19, 19, -w^{3} + w^{2} + 4w - 2]$ | $\phantom{-}\frac{704}{1609}e^{11} + \frac{1291}{1609}e^{10} - \frac{17047}{1609}e^{9} - \frac{54645}{3218}e^{8} + \frac{299011}{3218}e^{7} + \frac{414253}{3218}e^{6} - \frac{1145137}{3218}e^{5} - \frac{1313695}{3218}e^{4} + \frac{898993}{1609}e^{3} + \frac{746894}{1609}e^{2} - \frac{441117}{1609}e - \frac{460365}{3218}$ |
| 23 | $[23, 23, -w^{2} + 3]$ | $\phantom{-}\frac{314}{1609}e^{11} + \frac{2029}{6436}e^{10} - \frac{31675}{6436}e^{9} - \frac{45153}{6436}e^{8} + \frac{144487}{3218}e^{7} + \frac{363965}{6436}e^{6} - \frac{1148287}{6436}e^{5} - \frac{617775}{3218}e^{4} + \frac{1860795}{6436}e^{3} + \frac{751193}{3218}e^{2} - \frac{942755}{6436}e - \frac{479113}{6436}$ |
| 31 | $[31, 31, w^{3} - 4w + 2]$ | $-\frac{2891}{6436}e^{11} - \frac{4897}{6436}e^{10} + \frac{72589}{6436}e^{9} + \frac{27022}{1609}e^{8} - \frac{660655}{6436}e^{7} - \frac{858993}{6436}e^{6} + \frac{655578}{1609}e^{5} + \frac{2847903}{6436}e^{4} - \frac{1058822}{1609}e^{3} - \frac{3316079}{6436}e^{2} + \frac{2093021}{6436}e + \frac{530537}{3218}$ |
| 32 | $[32, 2, 2]$ | $-\frac{1905}{3218}e^{11} - \frac{1597}{1609}e^{10} + \frac{23230}{1609}e^{9} + \frac{33970}{1609}e^{8} - \frac{408585}{3218}e^{7} - \frac{521435}{3218}e^{6} + \frac{1557221}{3218}e^{5} + \frac{842523}{1609}e^{4} - \frac{1197717}{1609}e^{3} - \frac{978284}{1609}e^{2} + \frac{1129203}{3218}e + \frac{614797}{3218}$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $\phantom{-}\frac{553}{1609}e^{11} + \frac{2563}{3218}e^{10} - \frac{26091}{3218}e^{9} - \frac{53993}{3218}e^{8} + \frac{112425}{1609}e^{7} + \frac{402749}{3218}e^{6} - \frac{857499}{3218}e^{5} - \frac{622522}{1609}e^{4} + \frac{1367355}{3218}e^{3} + \frac{693002}{1609}e^{2} - \frac{675027}{3218}e - \frac{420467}{3218}$ |
| 53 | $[53, 53, -2w^{4} + w^{3} + 9w^{2} - 3w - 2]$ | $\phantom{-}\frac{1593}{6436}e^{11} + \frac{1044}{1609}e^{10} - \frac{17487}{3218}e^{9} - \frac{83431}{6436}e^{8} + \frac{279841}{6436}e^{7} + \frac{292139}{3218}e^{6} - \frac{1007989}{6436}e^{5} - \frac{1701125}{6436}e^{4} + \frac{1610107}{6436}e^{3} + \frac{1899271}{6436}e^{2} - \frac{423219}{3218}e - \frac{620761}{6436}$ |
| 59 | $[59, 59, -w^{4} + 5w^{2} + w - 4]$ | $-\frac{361}{1609}e^{11} - \frac{710}{1609}e^{10} + \frac{8762}{1609}e^{9} + \frac{15303}{1609}e^{8} - \frac{77136}{1609}e^{7} - \frac{117825}{1609}e^{6} + \frac{297022}{1609}e^{5} + \frac{376462}{1609}e^{4} - \frac{469288}{1609}e^{3} - \frac{424066}{1609}e^{2} + \frac{231763}{1609}e + \frac{131659}{1609}$ |
| 61 | $[61, 61, -w^{4} + w^{3} + 5w^{2} - 4w]$ | $-\frac{295}{1609}e^{11} - \frac{237}{1609}e^{10} + \frac{8622}{1609}e^{9} + \frac{7286}{1609}e^{8} - \frac{88889}{1609}e^{7} - \frac{76258}{1609}e^{6} + \frac{388028}{1609}e^{5} + \frac{312997}{1609}e^{4} - \frac{661504}{1609}e^{3} - \frac{419712}{1609}e^{2} + \frac{331950}{1609}e + \frac{151486}{1609}$ |
| 67 | $[67, 67, -w^{4} + 6w^{2} + 2w - 4]$ | $\phantom{-}\frac{131}{3218}e^{11} - \frac{201}{1609}e^{10} - \frac{2138}{1609}e^{9} + \frac{7613}{3218}e^{8} + \frac{44567}{3218}e^{7} - \frac{22535}{1609}e^{6} - \frac{180563}{3218}e^{5} + \frac{86639}{3218}e^{4} + \frac{248477}{3218}e^{3} - \frac{23733}{3218}e^{2} - \frac{39397}{1609}e - \frac{37659}{3218}$ |
| 71 | $[71, 71, 2w^{4} - w^{3} - 9w^{2} + 4w + 5]$ | $\phantom{-}\frac{3745}{3218}e^{11} + \frac{12917}{6436}e^{10} - \frac{185613}{6436}e^{9} - \frac{282103}{6436}e^{8} + \frac{415823}{1609}e^{7} + \frac{2225225}{6436}e^{6} - \frac{6475765}{6436}e^{5} - \frac{3695577}{3218}e^{4} + \frac{10204765}{6436}e^{3} + \frac{2209548}{1609}e^{2} - \frac{4955305}{6436}e - \frac{2873577}{6436}$ |
| 79 | $[79, 79, 2w^{4} - w^{3} - 10w^{2} + 2w + 7]$ | $\phantom{-}\frac{267}{6436}e^{11} + \frac{1109}{6436}e^{10} - \frac{3053}{6436}e^{9} - \frac{4433}{1609}e^{8} - \frac{3445}{6436}e^{7} + \frac{82661}{6436}e^{6} + \frac{29472}{1609}e^{5} - \frac{96503}{6436}e^{4} - \frac{73579}{1609}e^{3} - \frac{1109}{6436}e^{2} + \frac{194011}{6436}e + \frac{16445}{3218}$ |
| 83 | $[83, 83, -w^{4} + 2w^{3} + 5w^{2} - 7w - 2]$ | $-\frac{857}{3218}e^{11} - \frac{1583}{3218}e^{10} + \frac{20297}{3218}e^{9} + \frac{16152}{1609}e^{8} - \frac{174333}{3218}e^{7} - \frac{235177}{3218}e^{6} + \frac{329084}{1609}e^{5} + \frac{722497}{3218}e^{4} - \frac{515955}{1609}e^{3} - \frac{841533}{3218}e^{2} + \frac{492415}{3218}e + \frac{144695}{1609}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 3w + 3]$ | $-\frac{3013}{3218}e^{11} - \frac{3422}{1609}e^{10} + \frac{34693}{1609}e^{9} + \frac{140265}{3218}e^{8} - \frac{580957}{3218}e^{7} - \frac{508237}{1609}e^{6} + \frac{2148135}{3218}e^{5} + \frac{3062781}{3218}e^{4} - \frac{3336869}{3218}e^{3} - \frac{3386537}{3218}e^{2} + \frac{809591}{1609}e + \frac{1046365}{3218}$ |
| 83 | $[83, 83, w^{4} - w^{3} - 5w^{2} + 4w - 1]$ | $-\frac{173}{3218}e^{11} - \frac{1407}{6436}e^{10} + \frac{2733}{6436}e^{9} + \frac{19405}{6436}e^{8} + \frac{6615}{1609}e^{7} - \frac{57987}{6436}e^{6} - \frac{275577}{6436}e^{5} - \frac{19323}{1609}e^{4} + \frac{592117}{6436}e^{3} + \frac{47415}{1609}e^{2} - \frac{295087}{6436}e - \frac{74687}{6436}$ |
| 83 | $[83, 83, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ | $-\frac{2499}{6436}e^{11} - \frac{4233}{6436}e^{10} + \frac{62201}{6436}e^{9} + \frac{23358}{1609}e^{8} - \frac{558639}{6436}e^{7} - \frac{745901}{6436}e^{6} + \frac{543751}{1609}e^{5} + \frac{2507235}{6436}e^{4} - \frac{857438}{1609}e^{3} - \frac{3027123}{6436}e^{2} + \frac{1722281}{6436}e + \frac{501361}{3218}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ | $-1$ |